Damping Retention Three-Row Scan - Oscillations and Coupled Resonance Step by Step

Three exact retention rows show repeated multiplication without a continuous decay law. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Retention one loses no amplitude

Initial amplitude 8 m multiplied by retention 1 for 3 cycles leaves 8 m and loses 0 m.

A_n=8\left(1\right)^{3}=8\ \mathrm{m},\quad L=0\ \mathrm{m}

Half retention leaves one metre after three cycles

Initial amplitude 8 m multiplied by retention 1/2 for 3 cycles leaves 1 m and loses 7 m.

A_n=8\left({1\over 2}\right)^{3}=1\ \mathrm{m},\quad L=7\ \mathrm{m}

Three-quarter retention keeps a fractional remainder

Initial amplitude 8 m multiplied by retention 3/4 for 3 cycles leaves 27/8 m and loses 37/8 m.

A_n=8\left({3\over 4}\right)^{3}={27\over 8}\ \mathrm{m},\quad L={37\over 8}\ \mathrm{m}